An example showing that fibred quadratic polynomials admit many attracting curves

TL;DR

This paper constructs examples of fibred quadratic polynomials with multiple attracting invariant curves, highlighting a phenomenon impossible in non-fibred quadratic dynamics.

Contribution

It introduces the first known examples of fibred quadratic polynomials with multiple attracting invariant curves, demonstrating a novel dynamical behavior.

Findings

Existence of fibred quadratic polynomials with an attracting 2-curve

Construction of fibred quadratic polynomials with two attracting invariant curves

Such phenomena do not occur in classical non-fibred quadratic dynamics

Abstract

We present an example of a fibred quadratic polynomial admitting an attracting invariant 2-curve. By an unfolding construction we obtain an example of a fibred quadratic polynomial admitting two attracting invariant curves. This phenomena can not occur in the non-fibred setting.